This is the name of the Silverlode river, called Celebrant in Sindarin. The original name of the river was Buzundush "Darkroot", or Morthond in Sindarin. Tolkien later changed this to Zigil-nâd, but then decided to separate the concept of the color silver from the metal. He said that the Dwarves probably found silver in the river. At that point, the river's name changed one more time to Kibil-nâla, the form that made it into The Lord of the Rings.
Kibil is "silver, the metal". The Elves most likely derived the root celeb- from this word. Like kheled, it is probably singular in form, but can be used to denote silver as a material or uncountable noun. Here it would be in composition as the first element of the compound.
Nâla is more interesting. Tolkien says that "the meaning is not known. If it corresponds to -rant [in Celebrant] and -lode, it should mean 'path, course, river-course or bed' ". In English, a lode can be "a veinlike deposit, usually metalliferous" or, in British English, a "waterway or channel". This combination of definitions fits very well with the concept of this river. Looking to Semitic languages, we find the Arabic word wadi. It is described it as an intermittent stream or drainage course. I've read that a Hebrew word nahal, and Sanskrit nala are terms that are used synonymously with wadi. I've also seen that wadi might show up in Sanskrit instead as nad or nadi, which is of course close to the older term Tolkein used: Zigil-nâd. Seasonal or intermittent streams & rivers are found in the mountains as well as the desert, so I'm not surprised to see Khuzdul have a parallel term to wadi. I can see it being applied to rivers that always have water and/or perhaps have deep channels. This, I think, is the case for the rivers Narog and Ringwil in Beleriand of the First Age, at the confluence of which was the underground city of Nargothrond. I think then, that the Petty Dwarves' word for these caves, Nuluk-khizdîn, shows a form of nâla, nulu(k). See the discussion on Nuluk-khizdîn for further comment.
Nâla is thus probably a "path, course, bed, or channel of water or intermittent water" and singular, nominative, indefinite.
Kibil-nâla is "Silver Water-channel" (or similar) and is a compound word with noun-noun word order.
The Lord of the Rings, The Fellowship of the Ring, Book II, ch 3
The Treason of Isengard, pg 174-175
The Peoples of Middle-earth, pg 279, 286
Parma Eldalamberon XVII: Words, Phrases and Passages, pg 35, 36, 37